A family of sets ℱ is said to satisfy the (p, q) property if among any p sets in ℱ some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p ≥ q ≥ d + 1 there exists a minimum integer c = HDd(p, q), such that any finite family of convex sets in ℝd that satisfies the (p, q) property can be pierced by at most c points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on HDd(p, q), known as ‘the Hadwiger-Debrunner numbers’, is still a major open problem in discrete and computational geometry. The best known upper bound on the Hadwiger-Debrunner numbers in the plane is O(p(1.5+δ)(1+1q−2)) (for any δ > 0 and p ≥ q ≥ q0(δ)), obtained by combining the results of Keller, Smorodinsky and Tardos (2017) and of Rubin (2018). The best lower bound is HD2(p,q)=Ω(pqlog(pq)), obtained by Bukh, Matoušek and Nivasch more than 10 years ago. In this paper we improve the lower bound significantly by showing that HD2(p, q) ≥ p1+Ω(1/q). Furthermore, the bound is obtained by a family of lines and is tight for all families that have a bounded VC-dimension. Unlike previous bounds on the Hadwiger-Debrunner numbers, which mainly used the weak epsilon-net theorem, our bound stems from a surprising connection of the (p, q) problem to an old problem of Erdős on points in general position in the plane. We use a novel construction for Erdős’ problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on HD2(p, 3). We then generalize the bound to HD2(p, q) for q ≥ 3.
ASJC Scopus subject areas
- Mathematics (all)